All Questions
1,203 questions
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Map associated to linear system onto curve is morphism
In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
- 61
5
votes
0
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165
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Real endomorphism algebra of abelian surface is never $\mathbb{C}$?
I'm reading about the Sato Tate conjecture for genus 2, and I came to the paper here. This breaks the conjecture into 6 different parts, based on the real endomorphism algebra of the surface, or the ...
- 19.6k
5
votes
1
answer
627
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What is the maximal order of the automorphism group of a given Shimura variety?
Background:
Given an elliptic curve $E$, it seems that $max(ord(Aut(E)))$ over the prime 2 is 24, and $(max(ord(Aut(E)))$ over the prime 3 is 12.
The endomorphism algebra of an elliptic curve over $...
- 3,539
10
votes
1
answer
647
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$K_0$-equivalence of varieties
Let $k$ be an algebraically closed field of characteristic zero.
Let $A$ be the $\mathbf{Z}$-subalgebra of the Grothendieck ring of $k$-varieties $K_0(\text{Var}_k)$ generated by classes of semi-...
- 5,968
8
votes
1
answer
603
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absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$
In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{...
- 4,111
1
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1
answer
280
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Norms of elements in Artin-Schreier extensions
The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring":
Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ ...
- 17.6k
17
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2
answers
1k
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A short proof for simple connectedness of the projective line
The Riemann-Hurwitz formula implies that the projective line $\mathbb{P}^1_K$ over any algebraically closed field $K$ is simply connected (i.e., $\pi_1^{et}(\mathbb{P}^1_K) = 1$; equivalently, if $\...
CommunityBot
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2
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0
answers
146
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Are general ample divisors on abelian varieties smooth?
It is well known that in $\mathcal{A}_g$, the space of principally polarized abelian varieties, the general element has a smooth theta divisor (see Andreotti-Mayer, for example).
Now let $\mathcal{A}...
- 663
3
votes
0
answers
255
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What are the easiest counterexamples to Serre-Lang over non-algebraically closed fields?
Let $k$ be a field of characteristic zero. Let $A$ be an abelian variety over $k$. Let $X\to A$ be a finite etale morphism with $X$ a connected (smooth projective) variety over $k$.
Then, choosing a ...
- 161
10
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1
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552
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Orders of reductions of rational points on elliptic curves
I am looking for references where the following (or similar questions) have been studied:
Let $K$ be a number field or a function field in one variable over a finite field and let $E$ be an elliptic ...
- 13.8k
2
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0
answers
304
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Surjectivity of map of Picard schemes implies abelian
Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here.
I am looking for a reference or explanation of the fact that is used in Mumford'...
- 2,607
2
votes
0
answers
121
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Global invariant cycles in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. ...
- 728
6
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answers
231
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Faltings height variation "at place of bad reduction''
Is there any example in the literature where someone has considered the problem of bounding the variation of Faltings height at a place of bad reduction? Specifically, if $A_i$ for $i\in \{1,2\}$ are ...
CommunityBot
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328
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Connected component of Picard group of a genus $2$ curve and its Jacobian over an imperfect field
Let $H/k$ be a genus $2$ curve. Consider the function field of $H$ given by $k(H)$. Take the base extension of $H$ to $k(H)$, namely $\hat H:=H\otimes_k k(H)$. Consider the Jacobian $J$ of the curve $...
11
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1
answer
786
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A frustrating cohomology class on the moduli of abelian surfaces
Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...
- 40.3k
3
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0
answers
307
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Isotrivial factors of Jacobian
Let $k$ be an algebraically closed field of positive characteristic that it is not the algebraic closure of a finite field. Fix a smooth proper $k$-curve $C$ and write $J_C$ for its Jacobian abelian ...
- 728
14
votes
3
answers
979
views
Zeta function of Abelian variety over finite field
Let $A/\mathbf{F}_q$ be an Abelian variety of dimension $g$. Suppose one knows $|A(\mathbf{F}_{q^n})|$ for all $1 \leq n \leq g$. Does one know then $\zeta(A,s)$ (equivalently, $|A(\mathbf{F}_{q^n})|$ ...
- 156k
3
votes
0
answers
113
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Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$
Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
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0
answers
313
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Intersection of curves in abelian varieties
Fix an abelian variety $A$ of dimension at least $3$ and a (smooth, projective, irreducible) curve $C$ inside $A$ that generates $A$ as an algebraic group, over an algebraically closed field. Now, fix ...
- 30.5k
3
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0
answers
111
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Notation for theta divisors in _Tata Lectures on Theta II_
Is there anyone here who has worked through Mumford's Tata Lectures on Theta II and can help me with my difficulties in following what seems to be confusing and inconsistent notation?
I have had more ...
- 1,298
6
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1
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419
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independence of $\ell$ of characteristic polynomial of Frobenius on $\ell$-adic Tate module of Abelian varieties over number fields
I am looking for a reference for the independence of $\ell$ of the characteristic polynomial of the Frobenius $\mathrm{det}(1-|\kappa(v)|^{-s}\mathrm{Frob}_v \mid (V_\ell A)^{I_v})$ acting on the $\...
- 20.8k
14
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0
answers
1k
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A slick proof (?) of Zariski-Nagata purity in characteristic $p$
I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...
- 2,663
2
votes
0
answers
622
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How do I check if an abelian variety is principally polarized?
Let $V$ be a complex vector space of dimension $g$, and let $\Lambda\subseteq V$ be a full rank lattice endowed with a Riemann form $E\colon \Lambda\times\Lambda\to \mathbb Z$. Then the pair $(V/\...
- 509
2
votes
2
answers
1k
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About Weil's proof of "Weil conjectures for curves and abelian varieties"
I know that the Weil's proof of the Weil conjectures for curves and abelian varieties is made under the lenguage of his "Foundation of algebraic geometry", however in "Polarizations and Grothendieck's ...
CommunityBot
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2
votes
0
answers
124
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finiteness of Abelian varieties $B$ with $T_\ell A \cong T_\ell B$ for all primes $\ell$
Let $K$ be a number field.
In Faltings' Finiteness Theorems for Abelian Varieties over Number Fields, Corollary 3:
Let $A/K$ be an abelian variety, $d > 0$. Then there are only finitely many ...
CommunityBot
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4
votes
1
answer
807
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question regarding Faltings' proof of the Tate conjecture for Abelian varieties over number fields
Why does Faltings in his "Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern" in the proof of Theorem 3/4 assume that $W$ is a maximal isotropic $\pi$-invariant subspace? Tate also assumes ...
CommunityBot
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3
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0
answers
416
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The final step in the proof of Neron-Ogg-Shafarevich as in the paper of Serre-Tate
I have been reading the paper "Good Reduction of Abelian Varieties" of Serre-Tate and in particular the part where they show that the Tate module being unramified implies good reduction.
As in the ...
- 7,746
18
votes
1
answer
1k
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On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main result?
Let $k$ be a field, $\overline{k}$ its algebraic closure, and $A$ an abelian variety defined over $k$, of dimension $g$. For each integer $m \ge 1$, let $A_m$ denote the group of elements $a \in A(\...
CommunityBot
- 1
8
votes
0
answers
471
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Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k
6
votes
1
answer
250
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Are there curves of genus 2 with real multiplication by a non-maximal order?
Let us work over $\mathbb{C}$ for the moment.
Assume we are given a real quadratic field $K$ with ring of integers $\mathcal{O}_K$.
$\mathbf{Question:}$ Is there a smooth projective curve $C$ of ...
- 3,035
2
votes
1
answer
270
views
BSD conjecture for abelian schemes and the classical version
I would like to know the relation between the BSD conjecture for abelian schemes (as stated for example in T Keller's thesis) and the clasical BSD conjecture.
In particular, can one state the ...
- 50.8k
27
votes
7
answers
6k
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Etale covers of the affine line
In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
- 8,866
2
votes
0
answers
167
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The scheme structure on the Hilbert scheme of an Abel-Jacobi curve
Let $C$ be a smooth curve of genus $g\geq 3$, embedded in its Jacobian $X=\textrm{Jac } C$ via an Abel map. Let $\textrm{Hilb}_1(X)$ be the Hilbert scheme of curves in $X$, and let $[C]\in\textrm{...
- 430
3
votes
0
answers
231
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Automorphisms of Jacobians and polarizations
In a question previously asked on MO (Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication), a result from the paper ``On the fields of rationality for curves and for their ...
- 521
5
votes
1
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369
views
Do Abelian varieties isomorphic to all their conjugates descend?
Suppose $A$ is an abelian variety over $\overline{\mathbb{Q}}$ of dimension $g$, such that $A$ is isomorphic to all of its Galois conjugates. Note that I'm not including any polarization data.
Can I ...
- 2,665
23
votes
1
answer
1k
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Is hyperelliptic cryptography "practical"?
Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...
CommunityBot
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1
vote
1
answer
295
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Frobenius actions on cohomology of abelian variety
Let $A$ be an abelian variety over a finite field $k$. Let $V_\ell(A)$ be its $\ell$-adic Tate module.
We have a natural action of the absolute Galois group of $k$, and thus an action of the ...
CommunityBot
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7
votes
1
answer
812
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Galois action on $p$-adic Tate module of Abelian variety over finite field semisimple?
Let $A,B$ be positive dimensional Abelian varieties over a finite field and $p$ be an arbritrary prime. By Zarhin, Homomorphisms of abelian varieties over finite fields http://www.math.nyu.edu/~...
- 149k
2
votes
0
answers
121
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Direct factors of Jacobian
Is there any characterization of abelian varieties appearing as direct factor of the Jacobian of some curve?
Are there some special kind of abelian varieties that are known to be direct factor of ...
- 1,463
24
votes
5
answers
6k
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Wild Ramification
The question is, loosely put, what is known about wild ramification?
Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there ...
- 153
9
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1
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514
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Pull-back of an irreducible ample divisor via an isogeny of abelian varieties
In the (wonderful) book by C. Birkenhake and H. Lange Complex Abelian Varieties we can find the following result, see Corollary 4.3.4 page 77. It is stated in any dimension $g \geq 2$, but let us ...
- 66.3k
4
votes
0
answers
117
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How to describe the subspace of invariants under the Rosati involution?
Consider the Jacobian $J_C$ of hyperelliptic curve
$$C\!: y^2 = x^5 + a$$
over a finite field $\mathbb{F}_p$, where $a \in \mathbb{F}_p^*$, $p \equiv 2 \ (\mathrm{mod} \ 5)$, $p > 2$. Let $\pi \...
- 1,833
6
votes
1
answer
537
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Rationality of the Tate module of an abelian variety relative to the algebra of its endomorphisms
Suppose that $K/\mathbb{Q}_p$ is a finite extension and $k_K$ the residue field of $K$. Let $A/K$ be an abelian variety with good reduction. Suppose that $E\to\mathrm{End}^0_K(A)$ is an inclusion of a ...
- 149k
5
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0
answers
562
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Reduction of torsion points on Neron Model
Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $R$ and residue field $k$. Let $A/K$ be an abelian variety with Neron model $\mathcal{A}/R$. We denote by $\tilde{\mathcal{A}}/k$ the ...
- 421
1
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0
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348
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rigid analytic geometry positive characteristic
I am a beginning graduate student. I have the following basic question I am very confused about:
Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
- 147
21
votes
2
answers
5k
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State of resolution in positive characteristic?
Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these papers:
Kawanoue, Hiraku, Toward resolution of singularities over ...
- 7,800
1
vote
0
answers
162
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Theta functions, a natural basis.
Let $L$ be the principal polarization of an abelian variety $A$. Some people say that the theta functions with characteristics form a natural basis of the global section $\Gamma(A,L^{\otimes n})$ for $...
- 11
3
votes
1
answer
410
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Derivations of central extensions of simple Lie algebras
Let $L$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is not difficult to see (and also follows from Theorem 4.4 of [G. Hochschild: Semi-simple algebras and generalized ...
- 5,878
6
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0
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394
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Hrushovski's proof of the Manin-Mumford Conjecture
For my master's thesis, I am studying Hrushovski's model-theoretic proof of the Manin-Mumford Conjecture. Among the references I have used are the following:
Lecture notes 'Model Theory of Difference ...
- 61
17
votes
2
answers
1k
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Higher level analogs of Nicolas-Serre theory
NICOLAS-SERRE THEORY
Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke ...
- 5,422